Editing Structural proof theory (section)

==Structures and connectives<!--'Display logic' redirects here-->==
The term ''structure'' in structural proof theory comes from a technical notion introduced in the sequent calculus: the sequent calculus represents the judgement made at any stage of an inference using special, extra-logical operators called structural operators: in <math>A_1, \dots, A_m \vdash B_1, \dots, B_n</math>, the commas to the left of the [[Turnstile (symbol)|turnstile]] are operators normally interpreted as conjunctions, those to the right as disjunctions, whilst the turnstile symbol itself is interpreted as an implication. However, it is important to note that there is a fundamental difference in behaviour between these operators and the [[logical connective]]s they are interpreted by in the sequent calculus: the structural operators are used in every rule of the calculus, and are not considered when asking whether the subformula property applies. Furthermore, the logical rules go one way only: logical structure is introduced by logical rules, and cannot be eliminated once created, while structural operators can be introduced and eliminated in the course of a derivation.

The idea of looking at the syntactic features of sequents as special, non-logical operators is not old, and was forced by innovations in proof theory: when the structural operators are as simple as in Getzen's original sequent calculus there is little need to analyse them, but proof calculi of [[deep inference]] such as '''display logic'''<!--boldface per WP:R#PLA--> (introduced by [[Nuel Belnap]] in 1982)<ref>N. D. Belnap. "Display Logic." ''Journal of Philosophical Logic'', '''11'''(4), 375–417, 1982.</ref> support structural operators as complex as the logical connectives, and demand sophisticated treatment.